Highly symmetric lines

TL;DR

This paper introduces a unified approach to constructing highly symmetric line systems and frames using projective stabilizers, leading to new configurations that improve bounds on the kissing number in certain dimensions.

Contribution

It generalizes highly symmetric frames via projective stabilizers and constructs new line systems, including three new kissing configurations that enhance lower bounds on the kissing number.

Findings

Constructed new kissing configurations in dimensions 10, 11, and 14.

Improved lower bounds on the kissing number in these dimensions.

Provided examples illustrating the unified approach to symmetric line systems.

Abstract

A generalization of highly symmetric frames is presented by considering also projective stabilizers of frame vectors. This allows construction of highly symmetric line systems and study of highly symmetric frames in a more unified manner. Construction of highly symmetric line systems involves computation of twisted spherical functions associated with finite groups. Further generalizations include definition of highly symmetric systems of subspaces. We give several examples which illustrate our approach including 3 new kissing configurations which improve lower bounds on the kissing number in $d=10,11,14$ to 510, 592 and 1932 respectively.